We propose to study the following statistical problems arising in cancer studies. The results from this research will be useful in analyzing survival data. Median and Quantile Regression Models for Censored Data We will develop efficient numerical aIgorithms to implement the methods proposed by Ying, Jung and Wei (1993) for the median and quantile regression models with censored data; study the case when censoring variables depend on covariates; generalize these methods to multivariate failure time data; explore quantile regression methods for highly clustered failure time observations; and study model diagnostic procedures for the quantile regression model with censored data. B. Semi-parametric Methods for the Accelerated Failure Time Model We will study a class of new estimation procedures which can be obtained through standard numerical methods; carefully examine their performance and compare them to the rank estimators proposed by Wei, Ying and Lin (1990); derive confidence bands for the survival function under the accelerated failure time (AFT) model for future patients with a specific set of covariates; and study model diagnostic methods for the AFT model. C. Estimating the Difference Between Two Survival Curves We will study nonparametric methods for constructing simultaneous confidence intervals of the difference between two survival functions and explore appropriate transformations of the Kaplan-Meier estimates to obtain accurate bands for small sample sizes. We will study nonparametric methods which provide a single statistic for estimating and summarizing the difference between two survival curves.